Application of Maximum Entropy/Optimal Projection Design ...dsbaero/library/... · x(t) = Ax(t) (1)...

JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICSVol. 15, No. 5, September-October 1992

Application of Maximum Entropy/Optimal Projection DesignSynthesis to a Benchmark Problem

Emmanuel G. Collins Jr.* and James A. KingtHarris Corporation, Melbourne, Florida 32901

andDennis S. Bernsteinf

University of Michigan, Ann Arbor, Michigan 48109

Maximum entropy/optimal projection design synthesis is a methodology for designing robust, fixed-ordercontrollers for flexible structures. This paper reviews the theoretical basis for this method and illustrates theapproach using a benchmark problem. The benchmark problem involves two masses with spring coupling, anuncertain spring constant, and a sensor and actuator that are noncollocated. The results of this paper alsoillustrate the use of a precompensation methodology that allows the control designer to precondition the designplant by embedding judiciously designed filters. These filters are included in the implemented controller.

I. Introduction

A CTIVE feedback control for vibration suppression inlightly damped structures continues to be a challenging

area of aerospace engineering.1 Typically, such problems in-volve multi-input/multi-output systems in noncollocated sen-sor/actuator configurations. The task of designing distur-bance attenuation feedback controllers for such systems isfurther exacerbated by constraints on real-time processingcapacity for feedback control law implementation as well asmodeling uncertainty associated with complex structures.Among the numerous methodologies proposed for addressingthe structural control problem, this paper focuses on the max-imum entropy/optimal projection (ME/OP) approach. Thisapproach was originally developed by Hyland2'5 and Hylandand Madiwale6 in a series of technical reports and conferencepapers. The ME/OP approach has been applied experimen-tally to various structural control testbeds7'9 and has beenevaluated by other researchers in Refs. 10-12. Subsequently,numerous extensions and variations of this approach havebeen developed to address a variety of problems in robust,fixed-structure controller synthesis.13'17 A detailed review ofthis work with extensive references to related literature is givenin Ref. 18.

The purpose of this paper is twofold. First we provide (inSees. II and III) a brief review of robust fixed-order controllersynthesis, in particular, robust optimal projection controllersynthesis. Although a detailed review is given in Ref. 18, thebrief review given here focuses more directly on the ME/OPtechnique for robust controller synthesis, which is described inSec. III. After reviewing ME/OP, we then turn our attentionin Sec. IV to problems 1 and 2 of the benchmark examplegiven in Ref. 19. The ME/OP design approach is applied tothe benchmark example in Sec. V, and the design results arediscussed in Sec. VI. A conclusion is presented in Sec. VII.

II. Robust Fixed-Order Controller SynthesisOptimal projection theory13 generalizes linear quadratic

Gaussian (LQG) theory to the case of reduced-order con-

Received May 1, 1991; revision received Dec. 4, 1991; accepted forpublication Feb. 4, 1992. Copyright © 1992 by the American Instituteof Aeronautics and Astronautics, Inc. All rights reserved.

*Associate Principal Engineer, Government Aerospace Systems Di-vision. Member AIAA.

tSenior Engineer, Government Aerospace Systems Division.^Associate Professor, Department of Aerospace Engineering.

trollers. Although LQG theory provides quadratically (H2)optimal full-order dynamic compensators by means of twouncoupled Riccati equations, optimal projection theory char-acterizes quadratically optimal reduced-order (i.e., fixed-or-der) controllers via a coupled system consisting of two modi-fied Riccati equations and two modified Lyapunov equations.When the controller order is set equal to the plant order, theprojection matrix T responsible for the coupling (the so-called"optimal projection") becomes the identity, the couplingterms vanish, the Lyapunov equations are rendered superflu-ous, and the LQG Riccati equations are recovered. Numericalalgorithms for solving the optimal projection equations viaiterative and homotopy techniques are discused in Refs.20-22.

Robust optimal projection theory (as well as robust LQGtheory) has been developed by incorporating uncertaintybounds within the design optimization procedure. The ideabehind this approach can most clearly be illustrated within thecontext of robust analysis, whereas its application to synthesisis a fairly straightforward extension of fixed-structure opti-mization. The description given here follows the developmentof uncertainty bounds given in Ref. 23.

For the asymptotically stable linear system

x(t) = Ax(t) (1)

we consider a quadratic Lyapunov function of the form

(2)

where the positive-definite matrix P is given by the Lyapunovequation

0 = A TP + PA + R (3)

where R is positive definite. To address additive disturbancesfor a system of the form

x(t) = Ax(t) + w(/)

it is convenient to utilize the dual equation

0 = AQ + QAT + V

(4)

(5)

in which A is replaced by A T (which has the same spectrum asA) and where V is interpreted as the intensity of the whitenoise disturbance w ( t ) . In Eq. (5), the matrix Q can be viewed

1094

COLLINS ET AL.: MAXIMUM ENTROPY/OPTIMAL PROJECTION 1095

as a controllability Gramian or covariance matrix with associ-ated quadratic (H2) performance measure

J = tr QR = tr PV (6)

If A is uncertain so that Eq. (1) is replaced by

x(t) = (A + &A)x(t) (7)

where A A 6 11, a set of perturbations, then we wish to deter-mine whether A + AA remains stable for all AA € 11. Oneapproach to this problem involves replacing Eq. (5) by

0 = AQ + QAT + fi + V (8)

where 0 is a constant positive-definite matrix. Rewriting Eq.(8) as

= 04 + AA)Q

it follows that A + A A is stable so long as A A satisfies

(9)

<ti (10)

where Q is the solution to Eq. (8).A variation on Eq. (8) involves allowing 0 to be a function

of Q . Thus we consider the modified Lyapunov equation

Q = AQ + QAT +$l(Q)+ V (11)where 0( • ) satisfies

AAQ + QAA T < fi(Q), for all AA € 11 (12)

and for all non-negative-definite Q . It then follows by rewrit-ing Eq. (11) as

0 = 04 + AA)Q + Q(A + AA)T+ti(Q)

- (AAQ + QAAT)+ V (13)

that A + AA is stable. Furthermore, letting QAA satisfy

0 = 0 4 +AA)Q±A + Q±A(A + AA)T + V (14)

and subtracting Eq. (14) from Eq. (13) yields

0 = (X + AA)(Q - QLA) + (Q - Q±A)(A + AA)T

(15)which implies that

QAA < Q , AAt 11 (16)

Thus tr(QR) provides a worst-case bound for the actual H2performance tr(QAAR).

Since the ordering induced by the cone of non-negative-def-inite matrices is only a partial ordering, it should not beexpected that there exists an operator Q( - ) satisfying Eq. (12)that is a least upper bound. Indeed, there are many alternativedefinitions for the bound 12 ( • ). To illustrate some of thesealternatives, assume for convenience that A A is of the form

A A = (17)

where a\ is an uncertain real scalar parameter assumed only tosatisfy the stated bounds and A\ is a known matrix denotingthe structure of the parametric uncertainty. The bound 12( • )utilized in Ref. 24 for full-state-feedback design was chosen tobe the absolute value bound

(18)

where 1 • I denotes the non-negative-definite matrix obtainedby replacing each eigenvalue by its absolute value. Since thebound defined in Eq. (18) is not differentiable with respect toQ, it has limited usefulness in fixed-structure controller syn-thesis. A more useful bound is the quadratic (in Q) bound

(19)

which has been considered in Refs. 25 and 26. In Eq. (19), ALand AR are a factorization of A\ of the form A\ - ALAR. Athird bound that has been considered is the linear (in Q)bound

(20)

where a. is an arbitrary positive scalar. As discussed in Ref. 14,the linear bound is closely related to a multiplicative whitenoise model.

Within the context of fixed-structure controller synthesis,the linear and quadratic bounds have been merged with opti-mal projection theory in Refs. 15 and 16, respectively. Thequadratic bound also has the useful property that it enforcesan //«, (bounded real) constraint. This extension has beenincorporated within optimal projection theory in Ref. 17.

In summary, it can be seen that both the linear andquadratic bounds guarantee robust stability and performancewith respect to parameter uncertainty and lead to generaliza-tions of LQG and optimal projection theory. As discussed inRef. 27, however, these bounds actually guarantee robustnesswith respect to time-varying parameter variations, which maylead to conservatism when the parameter variations are knownto be constant. Viewed in the frequency domain, such boundscorrespond to small-gain-type conditions that enforce robuststability with respect to complex, frequency-dependent uncer-tainty, which is conservative if the uncertain parameters areknown to be real and constant. Such conservatism may haveserious consequences in controlling flexible structures withstiffness uncertainty, which is a highly structured, inherentlyreal form of parameter uncertainty. Consequently, we nowturn our attention to the maximum entropy/optimal projec-tion approach to robust controller synthesis that seeks toovercome these difficulties for a particular form of parametricuncertainty.

III. Maximum Entropy/Optimal ProjectionDesign Synthesis

As discussed in Sec. I, the ME/OP approach was originallydeveloped in Refs. 2-6. In brief, the basis of the ME/OP ideais to choose the operator fl(Q) in the modified Lyapunov equa-tion (8) to be of the form

0(Q) = (21)

where the summation corresponds to an uncertainty model ofthe form A + AA , where

AA = (22)

where a\,... ,or are uncertain real parameters, and d\,... ,6r> 0 are uncertainty scalings. Note that in Eq. (17) we set r - 1for convenience, although Eqs. (17-20) could readily be gener-alized to the case r > 1.

The unusual feature of Eq. (21) is that (as will be seenshortly) 0(Q) is not a bound in the sense of Eq. (12) as areEqs. (18-20). Thus we do not stipulate a precise uncertaintyrange for the uncertain parameters a/ as in Eq. (17). Rather,the constants d\,...,dr should only be viewed as scalings.Indeed, whereas the bounds of Eqs. (18-20) are valid forarbitrary choices of A\9 the operator in Eq. (21) will only beused (in this paper) under restrictive, but practically useful,assumptions. Specifically, we now assume that the nominal

1096 COLLINS ET AL.: MAXIMUM ENTROPY/OPTIMAL PROJECTION

dynamics matrix A is of the form

A = block-diag (23)

which is representative of a lightly damped structure in amodal basis, whereas the uncertainty matrix At is of the form

Af = block-diag ( 0,... , 0 , 1 ° * ,0,.. . ,0 j (24)

where the position of the matrix

0 1-1 0

corresponds to the /th diagonal block of A . Since the poles ofA + E/= j ffjAj are of the form - 77 + y (<*>/ + a/), each term a/^4,represents uncertainty in the imaginary part of a pole location.

To further illustrate the structure of !}(g) given by Eq. (21),define

so that Eq. (21) becomes

Q(Q) = SQ + QS + £ MiCM /*/ = i

(25)

(26)

and the modified Lyapunov equation (11) is of the formr

0 = (A + S)Q + Q(,4 + S)r+ £M/(M f + K (27)

Using Eq. (24) and the fact that

0 'N-i »J LS can be written as

It is interesting to contrast Eq. (26) with the linear bound ofEq. (20) in light of the structure of S. To do this, generalizeEq. (20) to the case r > 1 (but setting a: = 1), which now hasthe form

(29)

(30)

Now rewrite Eq. (29) as

where

4 1/2 £ 6/7/ = i

which yields a modified Lyapunov equation of the form

0 = (4 + S)Q + Q(A + S)7 + EM/CM/7* V (31)/ = i

which is identical to Eq. (27) with S replaced by S.Maximum entropy design appears to be closely related to

recent results on parameter-dependent Lyapunov functions

and variations of the Popov criteria.28 In Ref. 29, it is shownthat the maximum entropy covariance equation (31) describesthe average covariance for a Cauchy uncertainty probabilitydistribution.

Given the modified Lyapunov equation (27), the ME/OPdesign equations can be derived in a straightforward mannerfollowing the technique given in Ref. 15. Hence, consider thenominal plant

Bu +Awi , xtRns, u£Rn», WitRn»\ (32)

(33)

(34)

where y is the sensor output, z is the performance variable,and Wi and w2 are (for convenience only) uncorrelated, whitenoise disturbances with intensities V\ > 0 and V2 > 0, respec-tively. Also, consider the H2 cost functional

J(AC,BC,CC) = lim E(xTR{x + uTR2u) (35)

where R{ = E^El and R2 > 0. The matrices Ac, Bc, and Cccharacterize the A2cth-order dynamic compensator (nc < n)

xc = Acxc + Bcy, xc£Rnc

u = -Ccxc

(36)

(37)

Optimization of the performance functional (35) with themodified covariance model (27) applied to the closed-loopsystem yields dynamic compensator gains

c = Y(As-Qt-l,P)GT

Bc=TQCTV2~l

CC=R21BTPGT

(38)

(39)

(40)

where the n x n non-negative-definite matrices Q, P, Q, and Psatisfy

0 = ASQ + QAJ

0 = AjP

(41)

(42)

+ 6(^5 - EP)r+ G^C - r^GEGr7: (43)

0 = (As - Qt)TP + P(AS - Ql) + PEP - r^PEPr± (44)

rank g = rank P = rank QP = wc (45)

(46)

where ( • )# denotes the group generalized inverse and r has thefactorization

(47)

(48)

The matrix r defined in Eq. (46) is an oblique projectionmatrix (the optimal projection) that is responsible for enforc-ing the reduced-order constraint nc < n on the compensator.


The ME/OP design equations (41-46) can be solved byusing a homotopy algorithm.21'30 As illustrated by Fig. 1, thishomotopy algorithm allows the deformation of an LQG con-troller into a full-order maximum entropy controller. Themaximum entropy controller is then reduced to an appropriateorder by using an indirect controller reduction method. It isimportant that this initial reduced-order controller approx-imately solves the ME/OP design equations to within a smallerror, although it is not required to be a stabilizing controller.This can be achieved by beginning with a low authority LQGdesign and/or incorporating a sufficiently high level of uncer-tainty in the ME design. In practice a slight modification ofthe balanced controller reduction algorithm of Yousuff andSkelton31 is currently used as the indirect controller method.Once this initial reduced-order controller is obtained, the ho-motopy algorithm is used to deform this controller into anME/OP controller. Then, if a higher authority controller isdesired, the final step of the algorithm is to increase thecontroller authority to a desirable level.

IV. Benchmark ProblemWe now turn our attention to the benchmark problem of

Ref. 19. The dynamical system is described, and two associ-ated design problems are presented.

Consider the two-mass/spring system shown in Fig. 2,which is a generic model of an uncertain dynamical systemwith a noncollocated sensor and actuator pair. A control forceacts on body 1, and the position of body 2 is measured,resulting in a noncollocated control problem. This system canbe represented in state-space form as

(49)

00

— k/m\k/m2

00

k/m\- k/m2

1 0"0 10 00 0

00

1/m,0

u +

000

\/m2

DESIGN AN LQGCONTROLLER {OF POSSIBLY LOW

OR MODERATE AUTHORITY)"EASY PROBLEM"

USE HOMOTOPY ALGORITHMTO DEFORM LQG CONTROLLER

INTO A MAXIMUM ENTROPYCONTROLLER

USE AN INDIRECTMETHOD TO REDUCE

THE CONTROLLER ORDER"EASY PROBLEM"

USE HOMOTOPY ALGORITHMTO DEFORM CONTROLLER

INTO A MAXIMUM ENTROPY/OPTIMAL PROJECTION

CONTROLLER

USE HOMOTOPY ALGORITHMTO INCREASE CONTROLLER

AUTHORITY

Fig. 1 Practical application of the maximum entropy/optimal pro-jection control-design algorithm.

Fig. 2 Benchmark system.

y ' = z = x2, y = y f + vwhere

x\ =X2 =XT, =x4 =u =

Wj =z =

y ' =v =

(50)

position of body 1position of body 2velocity of body 1velocity of body 2control inputplant disturbanceperformance variable (output to be controlled)noise-free measurementsensor noise

Design ProblemsDesign Problem 1

Design a constant gain linear feedback compensator of theform

x = Acxc + Bcy

u = Ccxc + Dcy

(51)

(52)

(any of these matrices may of course be zero) with the follow-ing properties:

1) The closed-loop system is stable for m\ = m2 = 1 and0.5 < k < 2.0.

2) For w(t) = unit impulse at / = 0, the performance vari-able z has a settling time of about 15 s for the nominal systemmi = m2 = k = 1.

3) The control system can tolerate reasonable measurementnoise signals v(t).

4) Achieve reasonable performance/stability robustnesswith reasonable bandwidth.

5) Use reasonable controller effort.6) Use reasonable controller complexity.

Design Problem 2Same as design problem 1 except in place of step 2 insert the

following:The w(t) is a sinusoidal disturbance of frequency 0.5 rad/s

whose amplitude and phase, although constant, are not avail-able to the designer. Achieve asymptotic rejection of w(t) atthe performance variable z ( t ) [i.e., minimize lim sup^^^Owith a 20-s settling time] for m\ - m2 = 1, 0.5 < k < 2.0.

V. ME/OP Control Design for theBenchmark Problem

This section considers design problems 1 and 2 of the bench-mark problem described in the previous section. In particular,the development of robust controllers using the ME/OP ap-proach is described. This approach was first applied to thebenchmark problem in Refs. 32 and 33.

We begin by introducing some notation. Consider the plant

x(t) = Ax(t) + Bu(t)

y'(t) = Cx(t)

(53)

(54)

(55)

Then G(s) is said to be the transfer matrix representation ofEqs. (53-55) if

z(s)(56)


G(s)

Fig. 3 Design>configuration for the precompensation methodology.

w2(s)

H(s)

Fig. 4 Implementation configuration for the precompensation meth-odology.

Likewise (A , B, C, D},Ei) is said to be the state representationof Eq. (56) if Eqs. (53-55) hold.

Now, for a nominal value &nom of the spring stiffness let thecorresponding state representation of the benchmark systemshown in Fig. 2 be given by

x(t) = A0(knom)x(t) + B0u(t)

y'=z(t)

(57)

(58)

(59)

Also, let GO (s) be the transfer matrix representation of Eqs.(57-59).

A precompensation strategy was used for control law de-sign. This precompensation strategy is illustrated by Figs. 3and 4. As shown in Fig. 3 we simply embed the precompensa-tion filters CM(s), Cy'(s)9 CWl(s), and Cz(s) in the plant apriori and design the ME/OP controller H(s) for this modi-fied design plant. Then, as illustrated in Fig. 4, the precom-pensation dynamics Cu(s) and Cy>(s) are included in the im-plemented compensator H(s). The sensor noise w2(s) in Fig.3 is fictitious since it is added to the pseudo-output y'(s). Itsintensity V2 is chosen to aid in the determination of the con-troller authority. It is not difficult to show that for both Figs.3 and 4 the closed-loop transfer function Gd (s) satisfying

z(s)u(s) (60)

is identical in Figs. 3 and 4. Hence, this methodology insuresthat if H(s) is stabilizing in the feedback loop of Fig. 3, thenH(s) is stabilizing in the feedback loop of Fig. 4. In addition,Eq. (60) also insures that the transfer function between z(s)and Wi(s) is preserved, thus insuring the preservation of theattenuation from w\ to z. This precompensation methodologywas used in Ref. 9 to achieve controller roll off and to forcethe design plant to appear to be rate feedback. Its use for thebenchmark problem is detailed later.

To describe the control design process for each controller,assume that (A,B,C,D{, E{) is the state-space representationcorresponding to G(s) in Fig. 3, such that

(62)

(63)

(64)

is the state-space representation of the design plant. The syn-thesis of H(s) in Fig. 3 was based on the solution of the designequations (41-46).

The state-space basis of Eqs. (61-64) was chosen such thatA is block-diagonal with a 2 x 2 diagonal block of the form

0 o>0

corresponding to the vibrational mode of the system withnominal natural frequency co0. System uncertainty was as-sumed to be in the frequency of this mode, and thus the par-ameter r in Eqs. (41), (42), and (48) is given by r = 1. Thecorresponding uncertainty pattern matrix A j was given by

block-diag ( 0, . . . ,0, ^ ,0, . . . ,0 (65)

where location of the nonzero 2 x 2 block corresponded to thelocation of the dynamics of the vibrational mode in A . Thedesign weights R I , R2, V\, and V2 were given by

R, = (66)

x(t) = Ax(t) + Bu (/) (61)

The design parameters are p, which determines the controlauthority, and 5[/2, which weights A\ and reflects the level ofmodal uncertainty. Note that since the stiffness k was assumedto be in the interval [0.5 N/m, 2.0 N/m], the natural fre-quency co of the vibrational mode was in the interval [1.0rad/s, 2.0rad/s].

Three controllers are described next. Controllers 1 and 2were developed to meet the objectives of design problem 1whereas controller 3 was developed for design problem 2.Controllers 1 and 3 were formulated as standard H2 distur-bance rejection problems, i.e., Cu(s) = Cy>(s) = 1 in Figs. 3and 4. The disturbance weighting matrix CWl(s) was chosen inthe control design process for controller 3 to reflect knowledgeof the sinusoidal nature of the disturbance. For controller 2,precompensation Cu(s) was added to nullify the effects of therigid-body mode in a frequency band approximately onedecade above and one decade below the frequency of thevibrational mode. Basically, this was an attempt to make theproblem "easier" for the LQG part of the design. As will beshown, Cu(s) is simply a second-order lead-lag filter. The"lag" poles were included not only to make the precompensa-tion realizable in state space but also to prevent the LQGsegment of the controller from having to provide additionalroll off. The description of each controller includes the pre-compensation dynamics used to develop the design model andthe stiffness A:nom of the design model. The parameter knomwas not chosen to be 1 N/m as might be expected because itwas experimentally observed that as 6j increased the closed-loop system tended to become robust with respect to positiveperturbations in knom faster than with respect to negativeperturbations.

The settling time for each system was chosen to be the timerequired for the displacement of mass 2 to reach and staywithin the interval [-0.1 m, 0.1 m]. Each of the controllerssatisfied the corresponding settling time objectives when con-nected to the model corresponding to k = I N/m. Also, eachof the controllers stabilizes the plant for k € [0.5 N/m, 2.0N/m]. To illustrate the effectiveness of maximum entropydesign in inducing robustness, each of the three controllers iscompared with the LQG design that was used to initialize thedesign process illustrated by Fig. 1. The gain margin (GM) and


phase margin (PM) listed for each controller are the marginsyielded by implementing each controller with the correspond-ing design plant. In addition, for controllers 1 and 2 a simula-tion is provided that shows the mass 2 displacement responsewhen the sensor measurements are corrupted by a noise pro-cess w2(0- In these simulations, the noise was chosen to bewhite with a uniform distribution in the interval [-0.01 m,0.01 m]

Controller 1 (for Design Problem 1)The parameters for controller 1 are as follows:

Cu(s) = Cy(s) = CW{(s) = Cz(s) = 1

p = 0.00001, <5;/2 = 0.2

£nom = 0.6 N/m => o>0 = 1.0954 rad/s

order of G(s) = order of H(s) = 4 => full-order design

settling time of the mass 2 displacement = 15 s (for k = 1)

peak response of the mass 2 displacement = 0.7 m(for k = \)

IMPULSE RESPONSE AT MASS 2 FOR CONTROLLER 1 (K= 1.0.0.5.2.0)

H(s) = 194390(5 + 0.36679)[(5 - 0.11735)2 + 0.909962](s + 81.438)(s + 131.04) [(5 + 2.9049)2 + 1.86152]

Controller 1: stable for 0.45 < k < 2.05, GM - 3 dB,PM - 10 deg

0.3

°'2

10 15 20TIME (SECONDS)

IMPULSE RESPONSE AT MASS 2 FOR CONTROLLER 2 (K= 1.0,0.5.2.0)......... RESPONSE FOR K=0.5———— RESPONSE FOR K=1.0........ RESPONSE FOR K= 2.0


Fig. 5 Mass 2 displacement response to an impulse disturbance forcontroller 1 (top) and controller 2 (bottom).

IMPULSE RESPONSE AT MASS 1 FOR CONTROLLER 1 (K=1.0.0.5.2.0)• RESPONSE FOR K=0.5• RESPONSE FOR K=1.0

RESPONSE FOR K= 2.0


LQG controller: stable for 0.59 < k < 1.06, GM = 1 dB,PM < 1 deg

The impulse responses of the mass 2 displacement (the outputperformance variable), the mass 1 displacement, and the con-trol signal are shown, respectively, in Figs. 5-7 for k = 1.0,0.5, and 2.0 N/m. The impulse response of the displacementof mass 2 for k = 1.0 N/m with noise-corrupted measure-ments is shown in Fig. 8. The Nyquist plot of the loop transferfunction is shown in Fig. 9


IMPULSE RESPONSE AT MASS 1 FOR CONTROLLER 2 (K= 1.0.0.5.2.0)

Cu(s) =100[(5 + 0.04)2 + 0.06932]

(5 + 10)2 + 17.32052

p = 0.0001, a -0.2

knom = 0.6 N/m =» co0 - 1.0954 rad/s

order of G(s) = 6, order of H(s) = 4 => reduced-order design


peak response of the mass 2 displacement = 0.2 m(for k = 1)

......... RESPONSE FOR K=0.5———— RESPONSE FOR K=1.0........... RESPONSE FOR K= 2.0


25

Fig. 6 Mass 1 displacement response to an impulse disturbance forcontroller 1 (top) and controller 2 (bottom).

The impulse responses of the mass 2 displacement (the outputperformance variable), the mass 1 displacement, and the con-trol signal are shown, respectively, in Figs. 5-7 for k = 1.0,0.5, and 2.0 N/m. The impulse response of the displacementof mass 2 for k = 1.0 N/m with noise-corrupted measure-ments is shown in Fig. 8. The Nyquist plot of the loop transferfunction is shown in Fig. 10.


Cu(s) = - Cz(s) = 1, Cw.(s) =1

(5 + 0.00050)2 + 0.502

= 1.2247 rad/s

H(s) = 2490300(5 + 0.93838) [(5 + 0.30989)2 + 0.402372][(5 + 0.040000)2 + 0.0692822](s + 54.835)(s + 18.831)[(s + 10.000)2 + 17.3212][(s + 0.014561)2 + 0.0312412]

Controller 2: stable for 0.12 < k < 2.03, GM - 6 dB,PM = 33 deg

LQG controller: stable for 0.45 < kPM - 35 deg

1.35, G M - 7 dB,


£8CAfed

10

5

0

-5

-10

IMPULSE RESPONSES FOR CONTROLLER 1 (K=1.0, 0.5, 2.0)

———— RESPONSE FOR K=l.O- - - - - - RESPONSE FOR K= 2.0

yO(l \j \r^\l \j \j \j V V V \/ V V V _10 15 20

TIME (SECONDS)

s500

S o£-500

0

• s ?. n i • • • ———— RESPONSE FOR K=1.0i 1; : ; ' ' ! '•! •"• ?; •*! -U ': il '. * ii ,: ; » ' « • " "" ' RESPONSE FOR K=2-0S^jJ! • ! ! : ; ' : ; ; ! i ' i ! H i ! i ' i i i ' ^ i : i ' ! i ; ! ' : : i ; ; i l i ; i ; ; ! i : i i ! ; ! ^ n J ! ^ A ^ A A / i A A /' A .'•. A A A •• :\ -\

jhi i l l ! ij j i; i! ii si f H H H ̂ * ' v v * ; ; ' ; ' ' ' ' '

5 10 15 20 25 3TIME (SECONDS)

Fig. 7 Control signal response to an impulse disturbance for con-troller 1 (top) and controller 2 (bottom).

M2 IMPULSE RESPONSE FOR CONTROLLER 1 WITH NOISE (K=l)

0.3

lo.0.1

0

-0.1


M2 IMPULSE RESPONSE FOR CONTROLLER 2 WITH NOISE (K=l)


25

Fig. 8 Mass 2 displacement response to an impulse disturbance withsensor noise for controller 1 (top) and controller 2 (bottom).

order of <5(s) = order of //(s) = 6 => full-order design


peak response of the mass 2 displacement = 4.4 m(for k = 1)

150

100

50

-50

-150

CONTROLLER 1 WITH NOMINAL PLANT (K=.6)

pdb |

..-27.75 db

Fig. 9 Nyquist plot of loop transfer function for controller 1.

200

150

100

50

0

-50

-100

-150

-200

CONTROLLER 2 WITH NOMINAL PLANT (K=.6)

- -91.7 db

Fig. 10 Nyquist plot of loop transfer function for controller 2.

M2 RESPONSE PERSISTENT DISTURBANCE (W=SIN(.5T)) CONTROLLER 3

wu

iQ

— RESPONSE FOR K=0.5— RESPONSE FOR K=1.0— RESPONSE FOR K= 2.0


25 30

Fig. 11 Mass 2 displacement response to a 0.5 rad/s sinusoidal dis-turbance for controller 3.

86816(5 + 0.12320)[(5 - 0.21281)2 + 0.963302][(5 + O.Q23916)2 + 0.424272]5 + 253.19)(s + 38.684)[(s + 2.5068)2 + 1.67762][(s + 0.0011218)2 + 0.501382]

Controller 3: stable for 0.48 < k < 2.50, GM - 5 dB,PM = 22 deg

LQG controller: stable for 0.43PM - 22 deg

< k < 0.78, GM = 4 dB,

The responses of the mass 2 displacement (the output perfor-mance variable), the mass 1 displacement, and the controlsignal to a sinusoidal disturbance of frequency of 0.5 rad/s areshown, respectively, in Figs. 11-13 for A: = 1.0, 0.5, and 2.0N/m. It should be noted that no attempt was made to reducethe order of this controller.

VI. Discussion of ResultsControllers 1 and 2 both satisfied the settling time objectives

of design problem 1 but differed significantly in their basic

structure and overall performance. One of the primary differ-ences between these two controllers is best illustrated by theNyquist diagrams presented in Figs. 9 and 10 of the corre-sponding loop transfer functions. In these figures it is seenthat both controllers placed the loop transfer function in thefirst quadrant just before the nominal frequency o>0- Thisinsured stability when 180 deg of phase lag was added due tothe undamped vibrational mode of the plant. However, de-spite this similarity, the route that the controllers took to placethe loop transfer function in the first quadrant was vastlydifferent. As seen in Fig. 9, controller 1 chose to force the looptransfer function to reach the first quadrant via the secondquadrant. This required substantial phase lag from the com-pensator. However, compensator gain was also needed toachieve performance. Hence, controller 1 included a complex


Ml RESPONSE PERSISTENT DISTURBANCE (W=SIN(.5T)) CONTROLLER 3

......... RESPONSE FOR K=0.5———— RESPONSE FOR K=1.0.......... RESPONSE FOR K= 2.0


30

Fig. 12 Mass 1 displacement response to a 0.5 rad/s sinusoidal dis-turbance for controller 3.

the use of a precompensation methodology that essentiallyallows the control designer to precondition the design plant byembedding in this plant judiciously designed filters. Thesefilters are then included in the implemented compensator. Theorder of the controller designed using the precompensationmethodology was reduced using optimal projection theory.

AcknowledgmentsThis work was supported by the Air Force Office of Scien-

tific Research under Contracts F49620-89-C-0011 and F49620-89-C-0029.

RESPONSE TO PERSISTENT DISTURBANCE (W=SIN(.5T)) CONTROLLER 3

I °"• -0.5

- RESPONSE FOR K*0.5- RESPONSE FOR K=1.0RESPONSE FOR Kx 2.0

0 5 10 15 20 25 30TIME (SECONDS)

Fig. 13 Control signal response to a 0.5 rad/s sinusoidal disturbancefor controller 3.

pair of nonminimum phase zeros to achieve both phase lagand increased gain. On the other hand, the precompensationchosen for controller 2 was such that, as shown in Fig. 10, theloop transfer function reached the first quadrant via the thirdand fourth quadrants. The desired lead and gain increaseswere accomplished via minimum phase compensator zeros.

Figure 5 compares the impulse response of mass 2 for con-trollers 1 and 2 whereas Fig. 6 compares the impulse responseof mass 1 for the two controllers. It is easily seen from thesefigures that controller 2 reduced the effects of the impulsedisturbance on the system much more than controller 1. Inaddition, it is seen in the data listed for each controller thatcontroller 2 has significantly higher gain and phase marginsthan controller 1, a feature that might be desirable in someapplications. However, Figure 7 reveals that the greater per-formance of controller 2 was achieved at the expense of muchgreater control authority, and Fig. 8 shows that, due to itshigher bandwidth, controller 2 yields a closed-loop system thatis much more sensitive to sensor noise. In fact, the controlauthority required by controller 2 would likely require anactuator with mass many times the total mass of the originalsystem. Controllers 1 and 3 avoided the use of an impracticalamount of actuator authority by the judicious use of nonmin-imum phase zeros. We conjecture that a low authority con-troller that meets the performance objectives cannot be de-signed for the benchmark problem without using nonmini-mum phase zeros.

It is interesting to note that the data for controller 2 showsthat the initiating LQG controller has larger gain and phasemargins than the robust controller but also has less robustnesswith respect to the uncertainty in the stiffness k. This providesan illustration of a control problem in which the gain andphase margins do not necessarily imply parametric robustness.

The data for each of the three controllers clearly show thatthe initializing LQG designs did not satisfy the robustnessrequirements with respect to the uncertainty in k. The efficacyof maximum entropy design in providing the needed robust-ness was thus clearly illustrated by the results of this paper.

VII. ConclusionThis paper has demonstrated the application of the maxi-

mum entropy/optimal projection methodology for designingrobust reduced-order controllers to a benchmark problem.The use of maximum entropy uncertainty modeling clearlyprovided the needed robustness that was not achievable bysimple LQG control. The results of this paper also illustrate

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